On the Periods of Biperiodic Fibonacci and Biperiodic Lucas Numbers

Abstract

This paper is concerned with periods of Biperiodic Fibonacci and Biperiodic Lucas sequences taken as modulo prime and prime power. By using Fermat’s little theorem, quadratic reciprocity, many results are obtained.

1. Introduction

Fibonacci sequence and Lucas sequence are well-known sequences among integer sequences. The Fibonacci numbers satisfy the recurrence relation with the initial conditions and Binet’s Formula for the Fibonacci sequence iswhere and are roots of the characteristic equation . The positive root is known as “golden ratio.” Fibonacci numbers and their generalizations have many interesting properties and applications to almost every field of science and art [1–3]. In particular, Edson and Yayenie [4] introduced the Biperiodic Fibonacci sequence as follows:where and are two nonzero real numbers. We take and as integers. If we take , we get the Fibonacci sequence .

If we take , we get the Pell sequence . Similarly, if we take , we get the –Fibonacci sequence [5–8]. Binet’s Formula for the Biperiodic Fibonacci sequence is given aswhere are the roots of the characteristic equation and Moreover, This sequence and its properties can be found in [1, 4].

Another well-known sequence is the Lucas sequence which satisfies the same recurrence relation as the Fibonacci sequence with the initial conditions and Binet’s Formula for the Lucas sequence iswhere and are defined in (1). Bilgici defined generalization of Lucas sequence similar to the Biperiodic Fibonacci sequence as follows:where and are two nonzero real numbers. We take and as integers. This sequence and other generalizations of Lucas sequence with their properties can be found in [9, 10]. If we take , we get the Lucas sequence . Also, if we take , we get the –Lucas sequence [11].where , and are defined in (3).

On the other hand several researchers have made significant studies about the period of the recurrence sequences [2]. Wall [12] defined the period-length of the recurring series obtained by reducing a Fibonacci series by a modulus . As an example, the Fibonacci sequence isand has period 8. Vinson [3] and Robinson [13] both extended Wall’s study. Moreover, they studied the Fibonacci sequence for prime moduli and showed that for primes the period-length of the Fibonacci sequence divides , while for primes the period-length of the Fibonacci sequence divides

Gupta et al. [14] give alternative proofs of this results that also use the Fibonacci matrix. They place the roots of its characteristic polynomial in an appropriate splitting fields. Renault [15] examined the behaviour of the -Fibonacci sequence under a modulus.

Lucas studied the -Fibonacci sequence extensively. He assigned and deduced that if is quadratic residue (that is, a nonzero perfect square) , then If is quadratic nonresidue then Also, Rogers and Campbell studied the period of the Fibonacci sequence [16]. They investigated the Fibonacci sequence modulo prime and then generalized to prime powers.

2. Period of Biperiodic Fibonacci Sequence

In this section, we investigate the Biperiodic Fibonacci sequence modulo prime and then generalize to prime powers.

Definition 1. The period of the Biperiodic Fibonacci sequence modulo a positive integer is the smallest positive integer such thatBy the definition above, the only members that can possibly come back to the starting point are multiples of This can be summed up in the statement that if is the period of , then,

Theorem 2. Let be a prime and let be a positive integer. If then,

We remark that the proof of the Theorem 2 can be seen in [16].

Theorem 3. Let be a prime, let be a positive integer, and let and be the fundamental roots of the Biperiodic Fibonacci sequence. If is the period of ,

Proof. For and is a prime integer, we have and then Also we obtain If is even, then From , we getThus, From Theorem 2,If is odd,For and ,Therefore, . From Theorem 2,

Theorem 4. Let be an odd prime, let denote the period of , and let be a nonzero quadratic residue ; then

Proof. As is known from Fermat’s little theorem, Thus, we haveFrom , we have and then So that Therefore, (10) implies that

Lemma 5. If is a quadratic nonresidue , then

Lemma 6. Let and be the two roots of in . is a quadratic nonresidue ; then

Theorem 7. Let be an odd prime, let denote the period of , and let be a quadratic nonresidue ; then

Proof. From the Binomial theorem, we get It follows thatFrom Lemma 6, we obtainThus, Also, we haveThus, from (10),

Theorem 8. Let be a prime, let denote the period of , and let denote the period of If is even then and if is odd then

Proof. We have shown that in Theorem 3. So that If is even, thenFrom and by using Theorem 3, it follows thatThen,Thus, from (10),
If is odd thenSince and , then, Thus,

3. Period of Biperiodic Lucas Sequence

In this section, we investigate the Biperiodic Lucas sequence modulo a prime similar to Biperiodic Fibonacci sequence.

Definition 9. The period of the Biperiodic Lucas sequence modulo a positive integer is the smallest positive integer such that

For the same reasons as the Biperiodic Fibonacci sequence we have that if is the period of , then

Theorem 10. Let be an odd prime, let denote the period of , and let be a nonzero quadratic residue ; then

Proof. We use Fermat’s little theorem to get soWe know ; thusAlso, we haveBy using (10), we get

Theorem 11. Let be an odd prime, let denote the period of , and let be a quadratic nonresidue ; then

Proof. From Lemma 6 and (3), we get Thus, from (10),

Competing Interests

The authors declare that they have no competing interests.